### Stochastic Koopman Operators and DMD Algorithms Resistant to Noise

Oftentimes, complicated systems have a high degree of uncertainty; it can come from a lack of information on the system, noise in measurements, or true randomness in the evolution. Since a deterministic model fails to accurately represent the system, we can use the Stochastic Koopman Operator to predict the "expected value" of the future. When generating data-driven models of the system, standard Dynamic Mode Decomposition algorithms demonstrate a consistent bias. DMD can be shown to fail even for simple systems with a small amount of measurement noise in the data, With newer algorithms, we can generate models resistant to noise in data and randomness in the dynamics. We demonstrate a newer, more robust DMD algorithm which is resistant to noise in data. Additionally, it allows us to generate better models by introducing time shifts to the data similar to Hankel DMD, which fails when the system contains random elements.